Weighted power Maxwell distribution: Statistical inference and COVID-19 applications

During the course of this research, we came up with a brand new distribution that is superior; we then presented and analysed the mathematical properties of this distribution; finally, we assessed its fuzzy reliability function. Because the novel distribution provides a number of advantages, like the reality that its cumulative distribution function and probability density function both have a closed form, it is very useful in a wide range of disciplines that are related to data science. One of these fields is machine learning, which is a sub field of data science. We used both traditional methods and Bayesian methodologies in order to generate a large number of different estimates. A test setup might have been carried out to assess the effectiveness of both the classical and the Bayesian estimators. At last, three different sets of Covid-19 death analysis were done so that the effectiveness of the new model could be demonstrated.


Introduction
Covid-19 viruses, which have received a lot of attention over the last two years, are only one example of the massive waves of viruses we confront today. To play our part as statisticians in understanding and modelling the Covid-19 infections, It was necessary for us to devise a statistical model that could be used to fit and model the Covid-19 infections, despite the fact that these infections could be either continuous or discrete random variables. For the purpose of this study, we exerted a considerable amount of effort to select a model that offers a more satisfactory fit for the Covid-19 infections that were observed in a number of different countries. We compromised by introducing a new superior distribution as a blend of existing distributions in order to remedy the deficiencies of the standard distribution. In other words, we blended existing distributions together to create the new distribution. For more information, see [1][2][3][4][5][6][7][8][9]. One of the primary objectives of statistics is to develop applicable statistical models for occurrences in the real world that can be modelled using already existing probability distributions. This is one of the more important goals of statistics. In situations where the probabilities are being used to represent an event in life that could be harmful or unpredictable. Multiple probability distributions have been developed as a result of the complexity and difficulty of simulating real-life occurrences using standard distributions. This difficulty is a direct result of the fact that standard distributions are used. There are many different applications for these distributions to be used in. There are times when the probability distributions that are already known and easily accessible are unable to accurately reflect and describe the data for particular natural occurrences. This is due to the fact that these distributions are still in the process of developing. The generalised probability distributions end up being modified and expanded as a direct consequence of the changes and expansions brought about by this research. These modifications and expansions were brought about by this research.
The accuracy with which well-known probability distributions portray the tail shape of a distribution was improved by adding a few new or additional parameters to the models. This also improved the models' applicability to the data of real occurrences.
To show the dispersion of particles in thermal equilibrium, Maxwell (M) [10] developed a mathematical version of the Maxwell distribution. It has uni-modal and leptokurtic curves which makes it has the fact that not all molecules travel at the same speed. The Maxwell model, which has an impact on kinetic energy, explains many fundamental features of gases. It's also known as the momenta model, velocity model, degree of momenta model, and particle energy model, and it's used in astronomy, chemistry, and engineering. It was used as a lifetime distribution for the first time in [11], and Bayesian estimation for its parameter was provided. Chaturvedi and Rani [12] added new parameters to M model and derived Bayesian and non-Bayesian estimators. In recent research, several generalizations based on the M distribution have been presented and statistically verified. Its cumulative-distribution-function (CDF), and probability-density-function (PDF) are defined as follows where Section 8's data analysis begins. Last but not least, Section 9 contains the final observations that were extracted from this research paper.

Weighted power Maxwell distribution
To present the concept of a weighted distribution, suppose that X is a non-negative random variable with its PDF f(x), then the PDF of the weight random variable X w is given by where w(x) is a non-negative weight function and E(w(x)) = R x w(x)f(x)dx. Yadav et al. [17] derived the power Maxwell (PM) distribution with PDF defined as follows and its CDF is given by where E n ðzÞ ¼ R 1 1 e À zt t n dt, α > 0 is a scale parameter and β > 0 is a shape parameter. The c th moments of PM distribution is given by Let w(x) = x c , by using Eqs 3, 4 and 6, we have the PDF and CDF of WPM distribution as the following respectively, where E n ðzÞ ¼ R 1 1 e À zt t n dt. The concepts of dependability and hazard rate function are probabilities that define the lifetime (a random variable) from the beginning of a failings to the point where we must modify the production procedure. The life-time is measured from the beginning of a failings to the point where we must modify the production procedure. The terms dependability function and hazard rate function are used to refer to these probabilities, respectively. As a result of the incorporation of fuzzy factors into these functions, the scope of application for dependability and hazard rate functions has been broadened, which has resulted in a wider variety of applications that can make use of these functions. The fact that the colour red was chosen for the text indicates that traditional reliability models take into account true values, which means that they take into account the lifetime probability of the system component. However, in the real PLOS ONE world, this system lifespan accuracy is not accurate at all. This is due to the fact that the values of the system parameters obtained through experimentation, incorrect measurement are all susceptible to some degree of ambiguity. To put it another way, the accuracy of this statement is not accurate.
For the purposes of fuzzy reliability analysis, let T be a continuous random variable that reflects a system failure time (component). Calculating the fuzzy dependability requires only the fuzzy probability, which can be found in the formula [18].
where the membership function μ(x) indicates the extent to which each component of a given universe belongs to a fuzzy set. Assume now that μ(x) is It is possible to compute the lifespan of the fuzzy numbers function μ(x), which corresponds to a specific value, and this computation yields the result of θ − Cut, θ 2 [0, 1], can be obtained as: As a result, the values of fuzzy reliability could be calculated for all θ values. The fuzzy dependability of the WPM distribution is determined by the fuzzy reliability definition. Thus, the fuzzy reliability values of the WPM distribution can be obtained for all values of θ as S F ðtÞ 0<y<1 ¼ Z xðyÞ

Moments and moment generating function
The r th moments of the WPM distribution is given by  Fig 3 that as c increases, the mean and variance decrease, skewness and kurtosis increase. Also, as α increases, the mean and variance decrease, but skewness and kurtosis keep constant.
The r th incomplete moments of the WPM distribution is given by where G cþrþ3b 2b ; t 2b a � � is lower incomplete gamma function. The moment generating function of the WPM distribution is given by g is a hypergeometric function and see [19] for more reading.

Inequality curves
The Lorenz curve is the most extensively used and well-known of the inequality curves, and it has applications in a wide range of domains. It is defined for the WPM distribution as follows: where F(x p ) = p, I 1 (t) is first incomplete moments, x p is the quantile function and Also, we can determine Bonferroni and Zenga inequality curves according to their relationship with the Lorenz curve as the following (for more details see [20])

Entropies
This subsection contains different types of entropies for our proposed model such as Rényi, Tsallis, and Shannon entropies, for more details see [21][22][23]. The Rényi, K X (r), and Tsallis, L X (r) entropies of order r, where r > 0, r 6 ¼ 1 of our proposed distribution are given, respectively, by and

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The Shannon entropy of WPM distribution is determined as follows where FðzÞ ¼ d dz log½GðzÞ�.

Stress-strength reliability
Assume that X and Y are two independent random variables drawn from the WPM distribution with parameters (α, β, c 1 ) and (α, β, c 2 ), respectively. Then by using CDF 8 and PDF 7, we have stress-strength reliability of WPM distribution as follows x n Gða þ n þ 1Þ ;

Maximum likelihood estimation
Let x 1 , x 2 , . . ., x n be a random sample of size n from the PDF of the WPM model, then the likelihood function takes the form where Θ is the parameter vector (α, β, c). Then the log-likelihood function takes the form We have estimators of the parameters α, β and c of the proposed model via the MLE by differentiating Eq (16) with regard to α, β and c, respectively, and equating to zero.
and @'ðYÞ @c where ǴxðxÞ ¼ As it seems, from (17), (18) and (19), analytic solutions of MLEs of α, β and c are not available. Therefore, the "maxLik" package can be used numerically to perform an iterative Newton-Raphson (NR) approach in order to achieve the appropriate MLEsâ,b; andĉ for any given data sets. Once the maximum likelihood estimates of α, β and c calculated, the MLEs of the non-fuzzy reliability and fuzzy reliability indices S(t) (9) at any mission time t > 0 can be easily derived using the invariance property of MLEsâ;b andĉ aŝ

Asymptotic confidence intervals
To build the two-sided 100(1−γ)% ACIs for the unknown parameters α, β, and c, or any function of them such as S(t) and S F (t), say Θ = (α, β, c, S(t), S F (t)), the Fisher's information matrix Since the exact solutions of the Fisher's expectation is tedious to obtain, hence the asymptotic variance-covariance (V-C) matrix of the MLEsâ;b andĉ can be obtained by inverting IðaÞ and dropping E with replacing α, β and c by their MLEsâ;b andĉ, respectively, see [24]. Also, [25] stated that the MLEŶ and MPSEŶ are asymptotically analogous, in fact,Ŷ ¼Ŷ þ oðn À 1=2 Þ. However, by differentiating (16) partially with respect to α, β and c, locally at their MLEŝ a;b andĉ, the approximate V-C matrix, I À 1 ðâ;bĉÞ, is given by To construct the ACIs of S(t) and S F (t), one needs to obtain the corresponding estimated variances. Practically, the most statistically efficient method used to construct confidence intervals is called the delta approach. This approach is useful and easy to utilize compared to the empirically-driven bootstrap approach due to the latter should be taken as a second-choice alternative when the Taylor series approximation is empirically incorrect, see [26]. However, based on the asymptotic normality of the MLEs of the reliability parameters of life S(t) and S F (t), we haveŜðtÞ � NðŜðtÞ; s 2 SðtÞ Þ andŜ F ðtÞ � NðŜ F ðtÞ; s 2 S F ðtÞ Þ. According to the delta method, from (20), the ACIs for S(t) and S F (t) can be constructed using the corresponding normality, respectively aŝ where Δ S(t) and D S F ðtÞ are the gradient of S(t) and S F (t) obtained atâ;b andĉ as Hence, using the concept of large sample theory for MLEsŶ of Θ, the 100(1 − γ)% twosided ACIs for the unknown parameter α, β, c, S(t) and S F (t) is given bŷ whereŝ 2 Y is the estimated variance ofŶ, and z γ/2 is the percentile of the standard normal distribution with upper probability (γ/2) − th.

Bayesian estimators
The Bayesian paradigm has become the most common technique in several sectors in recent years, including but not limited to different applications. Its capacity to use prior information in the analysis makes it particularly valuable in dependability studies, where one of the major obstacles is data availability. The Bayes estimates and associated credible interval of the model parameters α, β, and c, as well as the fuzzy and non-fuzzy reliability parameters S(t) and S F (t), are discussed in this section.

Prior distribution and loss function
Because the gamma prior distribution can take on a variety of shapes depending on its parameter values, using separate gamma priors is a relatively simple process that can lead to discoveries with more explicit posterior density expressions. As a result, we evaluated gamma density priors, which are more adjustable in terms of altering support for the WPM distribution parameters than other difficult prior distributions. Thus, the WPM parameters α, β, and c are assumed to have independent gamma PDFs as Gamma(q 1 , w 1 ), Gamma(q 2 , w 2 ) and Gamma (q 3 , w 3 ), respectively. Then, the joint prior density of α, β, and c becomes where the hyperparameters q i , w i , i = 1, 2, 3, are chosen to reflect prior knowledge about the unknown parameters α, β and c, and they assumed to be known and non-negative, we determined the hyperparameters by eliciting hyper-parameters as El-Sherpieny et al. [27].
In literature, the choice of symmetric and asymmetric loss functions are an important issue in Bayesian analysis. So in this study for estimating the considered unknown quantities, the most widely-used symmetric loss function is the SEL function, Lð�Þ, which is defined as whereỸ being an estimate of Θ. Under (22), objective estimateỸ is given by the posterior mean of Θ. However, any other loss function can be easily incorporated.
The entropy loss function (ELF) is a good asymmetric loss function, according to Calabria and Pulcini [28]. The entropy loss function of the form is considered as whose minimum occurs atỸ ¼ Y. Then the Bayes estimator of Θ under entropy loss function isỸ

Posterior analysis
The joint posterior density function, π L (�) of α, β and c is given by where cjxÞdadbdc is the normalizing constant. Substituting (15) and (21) into (25), the joint posterior PDF of α, β and c becomes Under SELF (22) and ELF (24), the Bayesian estimator for any function of α, β and c, say Θ, is the posterior expectation of Θ. Therefore, it is necessary to acquire the marginal posterior distributions for each parameter of α, β, and c in order to develop these estimations, because (26) implicit mathematical expressions. It is quite evident that obtaining explicit forms for the marginal PDFs for each unknown parameter is not even remotely conceivable. In light of this, our intention is to use certain simulation techniques, such as the MCMC approaches, in order to generate the Bayesian estimates and the credible intervals that correspond to them.
First, from (26), we obtain the conditional posterior distributions of the supplied data for α, β, and c as follows: and From (27), (28) and (29), it is plainly clear that no known distribution can be analytically reduced to the conditional posterior distributions of parameters, and as a result, they cannot be sampled directly using the techniques that are generally accepted. Therefore, in order to mimic samples from (27), (28) and (29), the Metropolis-Hasting (MH) method using normal proposal distributions.
The MH technique is a very useful MCMC strategy since it can generate random samples from a posterior density distribution with an independent proposal distribution, as well as calculate Bayes estimates and generate HPD credible intervals. Furthermore, this technique provides an easy-to-apply chain form of the Bayesian estimate from a practical standpoint.

Simulations study and its numerical results
This section is devoted to exploring the performance of the introduced estimation methods for estimating the parameters of the proposed model using detailed simulation results. Several sample sizes, n = {50, 150}, and several values of the parameters, α = {0.25, 1.5}, β = {0.3, 1.3} and c = {0.6, 2}, are considered to generate N = 5000 random samples from the proposed distribution based on Eq (8). The average biases (Bias), and mean square errors (MSEs) are calculated. using the following equations The simulation results of the suggested model parameters using the nine estimation methodologies are shown in Tables 1-4. It is concluded from the Tables 1-4 that all of the suggested distribution's parameter estimates are extremely reliable and very close to their true values, with minimal biases, and MSEs in all circumstances. For all scenarios analysed, the proposed estimators are consistent, with the bias and MSE decreasing as n grows. The MLE and Bayesian estimate approaches based on symmetric and asymmetric loss functions perform admirably in estimating the proposed model parameters.

Modeling real data with analysis
In this section, we focus on three real data analyses which are relevant to Covid-19. The first data is taken from [29] and represents the United Kingdom transformed Covid-19 data and spans 82 days, from May 1, 2021 to July 16, 2021. For the first dataset, the transformation was performed by [29] according to the following procedure: where ND is daily new deaths, CC is daily cumulative cases, CD is daily cumulative deaths and x i is the transformed data. The first data is given as folllows:       reliability function have been successfully identified as a result of our investigation into its statistical features and determination of a linear representation for its PDF. For the purpose of obtaining point estimates for the unknown Weighted power Maxwell parameters α, β, and θ, several Bayesian and classical estimation approaches were taken into consideration. An R software was used to conduct a simulation study, which allowed for the comparison of the effectiveness of a number of different estimating strategies. To achieve this goal, the MCMC

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Weighted power maxwell distribution: Statistical inference and COVID-19 applications technique was used, and the results led us to the conclusion that the Bayesian approach is superior to all the traditional methods that were taken into consideration. We utilised a data collection called Covid-19 that was compiled in the United Kingdom. It was discovered that the Weighted power Maxwell distribution suited the data better than the majority of the other distributions that were being considered.

Future work
In the near future author we will make a full study on the vaccination of the patients and measure the rate of mortality after performing the vaccination an d we will try to make regression analysis to the future infection using new model.